“If I wanted to, I could also formulate syllogisms to respond to your sophistic reasoning, and better syllogisms than yours. But I reject such methods of argumentation and obtain my evidence from the Fathers and their writings. You shall answer me with Aristotle or Plato, or with one of your Modern Doctors. But to oppose you, I shall invoke the fishermen of Galilee, with their simple preachings and their true wisdom, which, to you, seem foolish.”
— Saint Symeon of Thessaloniki
Aristotle formulated a scheme of four causes — material, formal, agent and final — that could explain the current configuration of all things in the world. A well-known example of the applicability of this scheme is that of a table, in which its material cause would be the wood, its formal cause would be its design, its agent cause would be carpentry, and its final cause would be dining. In order to justify this scheme — that could be a sufficient explanation of motion (kinesis) in nature — Aristotle provided the concepts of potentiality (dunamis) and actuality (energeia). The former could be understood as the possibility or the capability that something has of becoming another, while the latter could be understood as something that is currently at work. This can be exemplified by a man who is currently a simple philosopher, but who has the potential to become the greatest philosopher of all time. In other words, before the greatest philosopher of all time was even born, a man and a woman had to exist and copulate, for the possibility of this philosopher to exist to be realised. The Greek philosopher then formulated that things are caused and everything that is caused is caused by another cause. And if everything that is caused is caused by another cause, and things are indeed caused, then either there is an infinite series of causes, or there is a first uncaused cause. Since an infinite series of causes presupposes circularity, one must assume that there is a first uncaused cause. Aristotle called this uncaused cause unmoved mover (Phys. VIII 4-6), although this is a spurious distinction since both concepts mean the same thing, a first uncreated and eternal thing that caused all others to exist. In Aristotle’s own words: “there is a substance which is eternal and unmovable and separate from sensible things. […] This substance cannot have any magnitude, but is without parts and indivisible” (Met. XII 7, 1073 a 3-5).
Throughout the history of philosophy, many scholars made different interpretations of the concept of uncaused cause. Perhaps the most prominent is that of the scholastics, that this first uncaused cause is a living, thinking and personal God.1placeholder On the other hand, there has always been an interpretation diametrically opposed to that of the scholastics, which has always rejected a monotheistic reading of Aristotle’s work.2placeholder I shall not provide a detailed analysis of these interpretations since this question — whether the concept of uncaused cause should be called God or not — has been the most asked question regarding causation. Rather, I shall focus on the most important question and, possibly, the sole one that holds any significance, that is, whether causation can be justified. The significance of this question lies in the fact that the whole system of philosophy of Aristotle is anchored in the principle of causation. A clear and succinct example of this is that scientific knowledge (sullogismon epistimonikon) — a knowledge that is given through demonstrations — is reliant upon syllogistic logic, since it regards the nature of demonstrations. Syllogistic logic, in turn, is reliant upon causation, since in a syllogism “the premises must be the causes of the conclusion, better known than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes” (APost. I 2, 71 b 28-32). Therefore, if causation cannot be justified, there can be no knowledge. Nonetheless, there are some different ways of approaching the question of whether causation can be justified. It is possible to provide an attempt of justifying causation through an ontological or epistemological discourse. However, many attempts have already been made through these forms of discourse which, ultimately, said little or nothing. A possible solution to avoid this philosophical palaver seems to be the development of a logic of causation. Hence the appropriate question to pose is whether causation can be justified from a logical point of view.
Chapter I — Deductive Logics
“The propositions of logic are tautologies. The propositions of logic therefore say nothing. […] Whereof one cannot speak, thereof one must be silent.”
— Ludwig Wittgenstein
1. What is the object of syllogistic logic?
Syllogistic logic, developed by Aristotle, is the earliest branch of formal logic. Overall, formal logic is concerned with reasoning and whether or not an argument is valid. Naturally, this also applies to syllogistic logic. However, the specificity of syllogistic logic lies in its exclusive concern with the form of the syllogism. Thus, in this section I shall focus on the nature of the object of syllogistic logic, i.e., the syllogism.
Arguments are traditionally divided into two types: deductive and inductive. The main type of argument in syllogistic logic is that made by deduction (sullogismos), which “is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so” (APrior. I 2, 24b18-20). In other words, the “things supposed” are premises, and the “results of necessity” are the conclusion. The study of deduction in syllogistic logic was based on the study of arguments that contained propositions of a specific type, the categorical propositions. The combination of two categorical premisses and a categorical conclusion result in a categorical syllogism. A categorical syllogism, in turn, is a specific type of deductive argument composed as follows:
Major premise — All M are P.
Minor premise — All S are M.
Conclusion — ∴ All S are P.
(Syllogism 1)
The terms P, S and M, each of which is used twice, are designated according to their function in the syllogism: the major term (P) is the predicate of the conclusion, the minor term (S) is the subject of the conclusion, and the middle term (M) connects the major term with the minor term. The structure of a categorical syllogism with its premises, conclusion and three terms, follows:
Major premise — Every vertebrate animal (M) has a spine and a skull (P).
Minor premise — Every mammal (S) is a vertebrate animal (M).
Conclusion — ∴ Every mammal (S) has a spine and a skull (P).
(Syllogism 2)
The logical form of a categorical syllogism is determined by its mood and figure. The mood determines the quantity and quality of the syllogism, i.e., whether each of its categorical propositions is universal or particular, and affirmative or negative (A, E, I or O). In Syl. 2., all categorical propositions are “A” propositions (of the form “all S are P”), therefore the mood of the above syllogism is “AAA”. The figure, in turn, determines the placement of the middle term (M) in both categorical premises. In the first figure, the middle term (M) is on the left in the major premise, and on the right in the minor premise. In the second figure, the middle term (M) is on the right in both premises. In the third, the middle term (M) is on the left in both premises. And in the fourth figure, the middle term (M) is on the right in the major premise, and on the left in the minor premise. There are 256 possible combinations of the forms of mood and figure, from which 15 forms of categorical syllogisms are unconditionally valid.3placeholder
Syllogistic logic consists in obtaining, through premises, the conclusive proof of the validity of an argument by means of necessity. This definition may serve as an approximate definition of deduction in general, given that syllogistic logic is essentially deductive. On account of that, what most logicians did for centuries with regard to the deductive logics was basically “to provide techniques for the appraisal of deductive arguments — that is, for discriminating between valid and invalid deductions” (Copi, 2019). Therefore, the validity of deductive arguments may or may not be asserted, depending on how they were structured, but the validity of deduction itself, as a pure method, is undoubted to most logicians.4placeholder Hence the unconditional validity of certain forms of syllogisms would be justified.
2. What is the object of formal logic?
A deductive argument is valid if, and only if, its premises entail its conclusion. Therefore, it is impossible for an argument to be valid if it is the case that its conclusion does not follow necessarily from its premises. In order to demonstrate that, I shall analyse the second example of the structure of a categorical syllogism given in §1. There are two things to be noted in Syllogism 2: (I) if the major and minor premises were true, the conclusion would also have to be true, and that would make the argument valid; (II) the validity of the argument has nothing to do with whether the premises are true or not, only that if it is that case that the premises are true, then the conclusion follows necessarily as true. Thus, if a deductive argument has a valid form, this is enough for the argument to be valid, and there is nothing else to be done from a purely logical point of view. It is for this reason that perfect syllogisms “are valid regardless of whether their terms denote actually existing things” (Hurley, 2011). That is, a deductive argument can be absolutely false and, even so, valid. For example:
Major premise — Every wolf-like creature (M) studies formal logic (P).
Minor premise — Every werewolf (S) is a wolf-like creature (M).
Conclusion — ∴ Every werewolf (S) studies formal logic (P).
(Syllogism 3)
All categorical propositions of the argument above are evidently false. However, the argument is still valid, for it is not necessary for an argument to have true premises or a true conclusion in order to be valid. The reason for this is because logic is solely and exclusively concerned with the special relation of entailment that an argument holds. For Łukasiewicz, author of a magnificent work on Aristotle’s syllogistic logic, in which he makes a new approach to it from the point of view of modern formal logic:
“Only syllogistic laws stated in variables belong to logic, and not their applications to concrete terms. The concrete terms, i.e., the values of the variables, are called the matter, ϋλη, of the syllogism. If you remove all concrete terms from a syllogism, replacing them by letters, you have removed the matter of the syllogism and what remains is called its form” (Łukasiewicz, 1957, p. 16).
On the other hand, if it is the case that apart from having a valid form, an argument also has true premises, then the argument is sound. A sound argument must necessarily meet two requirements: (I) be valid and (II) have true premises. Therefore, a distinction must be made between what is a valid argument and what is a sound argument, since there are arguments that are valid and unsound simultaneously. It must be noted that Syllogism 3 is a valid argument, although unsound — for it has a valid form, although none of its premises are true — and that Syllogism 2 is a sound argument — for it meets both requirements of a sound argument, i.e., it is a valid argument, and both of its premises are true. However, some serious consequences derive from this distinction between validity and soundness. For this reason, I shall analyse it in detail in the next sections. What can be concluded so far is that one cannot assign truth values to propositions in deductive logics, given that it regards only the form of arguments and the relation of entailment of these propositions. Therefore, it is not the task of pure logic to investigate the truth or falsity of propositions, but rather to establish the validity of arguments, from the places in which presumably true or false propositions are arranged in these arguments. Furthermore, pure logic cannot create anything new. If one removes the content or the value of variables, only the arrangement of these variables remains, hence the application of the laws of logic to reality is of no value to pure logic. Therefore, the exercise of pure logic is ultimately constrained to proceed towards tautologies.
Chapter II – Non-Deductive Logics
“This endless, tiresome juggling of concepts […], this useless kaleidoscope of abstract categories spinning ceaselessly before the mind’s eye, was bound in the end to blind it to those living convictions that lie above the sphere of rationalistic understanding and logic — convictions to which people do not attain through syllogisms, but whose truth, on the contrary, people can only distort, if not utterly destroy, through syllogistic deduction.”
— Ivan Kireyevsky
3. The problem within material logic
In order to exemplify the use of the concepts of validity and soundness, and relate it to the distinction between the form and the matter of an argument, I shall provide the formalisation of Aristotle’s argument for causation. In natural language, the argument consists of the fact that: things are caused (R) and everything that is caused is caused by another cause (Q). And if everything that is caused is caused by another cause (Q), and things are caused (R), then either there is an infinite series of causes (S), or there is a first uncaused cause (P). The formalisation and logical proof follow:
Φ = {R · Q, (Q · R) ⊃ (S ⊻ P), ¬S}
- R · Q (Premise)
- (Q · R) ⊃ (S ⊻ P) (Premise)
- ¬S (Premise)
- S ⊻ P (1, 2, MP)
- ∴ P (4, 3, DS)
Φ ⊢ P.5placeholder
(Formal proof 1)
The proposition P stands for “there is a first uncaused cause”. The logical proof I provided indicates that the proposition P is proved through the set of propositions Φ, therefore it is valid to conclude that there must be a first uncaused cause. This logical proof is more refined than the syllogisms seen previously, although they share the same underlying idea. This is evident due to the following statements: (I) the set of propositions Φ is a set of premises, and (II) the conclusion follows necessarily from this set of premises. From the point of view of validity, this logical proof is perfect. However, there is a serious problem in trying to establish the soundness of this argument. In reality, this problem is not unique to this particular argument but rather extends to all conceivable arguments.
There are two possible types of analysis which regard arguments: formal and material. The formal analysis consists of everything that has been established so far, that is, establishing the validity of arguments from their form. Hence it is not necessary to provide more examples of this type of analysis. On the other hand, the material analysis consists of the attempt to pre-establish the truth or falsity of propositions in a given argument. As an example, it was previously established that Syllogism 3 is a valid argument, although unsound, for its propositions are false. Since the truth or falsity of these propositions are of no value to the formal analysis within pure logic, a distinct type of analysis, known as material analysis, becomes necessary to pre-establish the truth or falsity of these propositions. However, it must be noted that, in material logic, one cannot eliminate the matter of the arguments and proceed towards conclusions by means of necessity. Rather, this logic is solely concerned with the truth of the premises and conclusions of an argument. The problem within material logic lies in the fact that it presupposes an intrinsic relation between critical thinking and the content of thoughts with empirical reality. That is, a barrier which does not belong to the domain of pure logic emerges: relating pure thought to the external world. However, I shall not delve into the implications associated with the numerous theories of truth that exist, as it would not align with the purpose of my essay.
It is enough to note that although Aristotle’s formalised argument is valid, it cannot be defined as sound unless material logic is justified. And from the moment a logic depends on a body of data from empirical reality for a conclusion to be derived, it can no longer be deductive. The central idea of this chapter revolves around the fact that this is precisely the modus operandi of material logic. That is, in order to establish the truth or falsity of a premise X in a given argument A, it is necessary to construct an argument B, whose premises provide evidential support for the conclusion that the premise X in the given argument A is either true or false. Thus, in order for material logic to hold any significance, there must exist a non-deductive logic whose operation precedes that of material logic and, simultaneously, can both justify it and be justified. This logic is referred as inductive logic.
4. The problem within inductive logic
Inductive logic is the study of inductive arguments. Inductive arguments were mentioned in §1, although not introduced, for deductive arguments are the foundational concern of syllogistic logic. One of the main differences between deductive and inductive arguments lies in the fact that deductive arguments must be evaluated in terms of validity, while inductive arguments, of probability.6placeholder The reason for this is that, in inductive logic, no conclusion can be derived from its premises by means of necessity. Rather, the conclusion always presupposes generalisation. As an example:
Premise — All swans ever observed are white swans.
Conclusion — ∴ All swans are white.
(Inductive argument 1)
The conclusion of the above argument derives from the generalisation of the content of the premise. When comparing the form of the syllogism 2 and the generalisation within the inductive argument 1, it is concluded that deductive arguments surpass inductive arguments, given their consistency, which is always reliant upon the idea of logical necessity. However, the problem of induction is not limited to its inherent inferiority when compared to deduction. Its consequences have been disastrous to any positive stance on knowledge, and has led to nothing other than scepticism. The problem within inductive logic lies in the fact that one cannot predict future events, and for this reason, one cannot justifiably infer that future events will resemble events observed in the past (Wittgenstein, 1921, propositions 6.363-6.37). As an example, the generalisation within the inductive argument 1 consists of, ultimately, presuming that future observations will always resemble past observations, i.e., that all swans will always be white, for all previously observed swans have been white. It must be noted that there are different methods of induction — enumerative, analogical, eliminative, probabilistic, etc. — although they all originate from the same defective principle. Furthermore, one cannot even assume certain inferences as being more probable than others, for this would still be reliant upon the idea that it is possible to predict the future from past observations (Hume, 1748, Book I, Part III, Section VI). In conclusion, there are no justified inductive inferences, for there is no non-circular justification of induction. Therefore, induction is unjustified.
Considering that the primary objective of my essay is to establish whether causation can be justified from a logical point of view, I provided the formal proof of Aristotle’s argument for causation in §3, in order to avoid the imprecision of natural language. As for the formal proof, there are three things to be noted: (I) the premises were presumed to be true, (II) given that logical proofs are deductive, they can only concern the relation of entailment that the arrangement of its variables hold, and (III) regardless of the matter of the argument, as long as the premises are presumed to be true, the formal proof will always be valid. Hence validity is related only to variables, not to the matter which the variables might represent. As for the idea of soundness of an argument, there is very little that can be said. Since the material analysis is reliant upon the general claim of induction, which, in turn, is unjustified, to establish whether a premise is true or not does not belong to the domain of logic. The same applies to the principle of causation. Given that causation is reliant upon the data of the external world, and that the relation of pure thought with the external world does not belong to the domain of logic, causation is illogical in any case. To the question whether causation can be justified from a logical point of view, the answer is no.
Abbreviations & Works Cited
(APrior) Prior Analytics.
(APost) Posterior Analytics.
(EN) Nicomachean Ethics.
(Met) Metaphysics.
(Phys) Physics.
Barnes, J. (1995). The Cambridge Companion to Aristotle. Cambridge: Cambridge University Press.
Barnes, J. (2001). Aristotle: A Very Short Introduction. Oxford: Oxford University Press.
Carlini, A. (1925). Aristotele: Il problema religioso (Lib. XII della Metafisica e Frammenti). Bari: Gius. Laterza & Figli Tipografi Editori Librai.
Copi, I. (2019). Introduction to Logic. New York: Routledge.
Guthrie, W. (1933). The Development of Aristotle’s Theology. Cambridge University Press. The Classical Quarterly.
Hurley, P. (2011). A Concise Introduction to Logic. Boston: Wadsworth Publishing.
Hume, D. (1739-1740). A Treatise of Human Nature. London: John Noon.
Hume, D. (1748). An Inquiry Concerning Human Understanding. London: A. Millar.
Kireyevsky, I. (1856). On the Necessity and Possibility of New Principles in Philosophy.
Łukasiewicz, J. (1977). La silogística de Aristóteles desde el punto de vista de la lógica formal moderna. Madrid: Editorial Tecnos.
Mugnier, R. (1930). La théorie du primier moteur et l’évoution de la pensée aristotélicienne. Paris: J. Vrin.
Reale, G. (2008). Il concetto di filosofia prima e l’unità della metafisica di Aristotele. Milan: Bompiani Editore.
Wittgenstein, L. (1921). Tractatus Logico-Philosophicus (C.K. Ogden, Trans.). London: Kegan Paul.
Cf. the “Summa Theologica” of St. Thomas Aquinas. London: Burns, Oates & Washburne, ltd; Carlini, A. (1925). Aristotele: Il problema religioso (Lib. XII della Metafisica e Frammenti). Bari: Gius. Laterza & Figli Tipografi Editori Librai; Reale, G. (2008). Il concetto di filosofia prima e l’unità della metafisica di Aristotele. Milan: Bompiani Editore.
Cf. Mugnier, R. (1930). La théorie du primier moteur et l’évoution de la pensée aristotélicienne. Paris: J. Vrin; Guthrie, W. (1933). The Development of Aristotle’s Theology. Cambridge University Press. The Classical Quarterly, vol. 27 (No. 3/4), pp. 162-171; Barnes, J. (1995). The Cambridge Companion to Aristotle; Barnes, J. (2001). Aristotle: A Very Short Introduction.
Cf. Łukasiewicz, J. (1977). La silogística de Aristóteles desde el punto de vista de la lógica formal moderna. Madrid: Editorial Tecnos.
The problem of induction is widely known in philosophy mainly by logicians and epistemologists, who for some centuries have been trying to deal with this thorn in their side. However, a much lesser-known problem — which is perhaps more important than the problem of induction — is that regarding the theory of deduction, which suggests that it lacks justification. The consequences of this problem are severe, and the existence of paraconsistent logic itself owes, in part, to the revision of this apparent unconditional validity that had always been attributed to deduction. However, I will certainly not be able to discuss this question in detail in the present writing, on account of its complexity. See: Carroll, L. (1895). What the Tortoise Said to Achilles. Mind IV (14), pp. 278-280; Quine, W. V. O. (1976). The Ways of Paradox, and Other Essays. Cambridge: Harvard University Press; Costa, N. C. A. (2008). Ensaio Sobre os Fundamentos da Lógica. São Paulo: Editora Hucitec; Başkent, C. and Ferguson, T. (2019). Graham Priest on Dialetheism and Paraconsistency. In: Outstanding Contributions to Logic, 18. New York: Springer Publisher.
Φ ⊢ P means Φ syntactically entails (proves) P. It must be noted that the premises of this proof were assumed to be true, and the third proposition — i.e., the last premise in the set Φ — is “it is not the case that there is an infinite series of causes”. The reason the negation of S is assumed as a premise is that if it were the case that there was a never-ending series of causes, nothing could exist. However, things exist. Therefore, it is not the case that there is a never-ending series of causes. This is a perfect example of a modus tollens: P→Q, ¬Q ∴ ¬P.
Cf. Henderson, L. (2022). The Problem of Induction. The Stanford Encyclopedia of Philosophy, Winter 2022 edition. URL: <https://plato.stanford.edu/archives/win2022/entries/induction-problem/>.